Risk-averse estimation, an axiomatic approach to inference, and Wallace-Freeman without MML

TL;DR

This paper introduces a new class of risk-averse Bayesian estimators, axiomatizes inference requirements, and uniquely characterizes estimators like MAP and Wallace-Freeman, providing a novel justification without approximations.

Contribution

It axiomatizes risk-averse Bayesian estimation, characterizes estimators like Wallace-Freeman, and offers a new, exact justification for Wallace-Freeman without relying on approximations.

Findings

Axioms uniquely characterize MAP and Wallace-Freeman estimators.

Provides a novel, exact justification for Wallace-Freeman estimator.

Connects estimation axioms with well-known Bayesian estimators.

Abstract

We define a new class of Bayesian point estimators, which we refer to as risk averse. Using this definition, we formulate axioms that provide natural requirements for inference, e.g. in a scientific setting, and show that for well-behaved estimation problems the axioms uniquely characterise an estimator. Namely, for estimation problems in which some parameter values have a positive posterior probability (such as, e.g., problems with a discrete hypothesis space), the axioms characterise Maximum A Posteriori (MAP) estimation, whereas elsewhere (such as in continuous estimation) they characterise the Wallace-Freeman estimator. Our results provide a novel justification for the Wallace-Freeman estimator, which previously was derived only as an approximation to the information-theoretic Strict Minimum Message Length estimator. By contrast, our derivation requires neither approximations nor coding.